SOLUTION: A population of values has a normal distribution with μ=36.5 and σ=79.4 .If a random sample of size n= 24 is selected, Find the probability that a sample of size n=24 is rando

Question 1186770: A population of values has a normal distribution with μ=36.5 and σ=79.4 .If a random sample of size n= 24 is selected,
Find the probability that a sample of size n=24 is randomly selected with a mean greater than 36.5. Round your answer to four decimals.
P(M > 36.5) =
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to find the probability:
1. **Understand the problem:** We're dealing with a normal distribution, and we want to find the probability that the sample mean (M) is greater than the population mean (μ).
2. **Central Limit Theorem:** The Central Limit Theorem states that the distribution of sample means will be approximately normal, even if the original population isn't perfectly normal, as long as the sample size is large enough (n ≥ 30 is a common rule of thumb, but n=24 is close enough). The mean of the sample means will be equal to the population mean (μ = 36.5).
3. **Probability:** Since the distribution of sample means is normal and centered at the population mean (36.5), the probability of the sample mean being greater than 36.5 is exactly 0.5 (50%). Half of the sample means will be above the population mean, and half will be below.
Therefore, P(M > 36.5) = 0.5000

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